|
|
||
|
168 VIII DIFFERENTIAL CALCULUS
|
||
|
|
||
|
Here again, to apply (8.5.1) we have only to verify that £ -* ||/(£)ll is regulated.
Finally, we express for integrals results corresponding to (8.6.4) and (8.6.5): |
||
|
|
||
|
(8.7,8) If a sequence (gn) of regulated functions, defined in a compact interval
I = [a, /?], converges uniformly in I to g, then the sequence y a gn(£) d£) con- verges to (^#(0 d%. (Remember g is regulated by (7.6.1).) |
||
|
|
||
|
(8.7.9) If a series («„) of regulated functions, defined in a compact interval
00
I = [a, /?]» & normally convergent (Section 7.1) in I, then, if u = ]T w«>
n = Q
I*B
the series of general term un(Qd£ is absolutely convergent, and
Jo.
|
||
|
|
||
|
The absolute convergence follows at once from the assumption and the
mean value theorem (8.7.7). |
||
|
|
||
|
(87.10) Remark. Due to (8.6.4) and the proof of (7.6.1), for any regulated
function /defined in [a, /?], and for any e > 0, there is an increasing sequence |
||
|
|
||
|
such that
n-l
|
||
|
|
||
|
Ja
|
||
|
|
||
|
If /is continuous, one may (due to (3.16.5)) take all numbers xfc+1 — xk
equal to (j3 - a)//i, and rfc = xk (see Problem 1). |
||
|
|
||
|
PROBLEMS
1. Let / be a regulated function defined in a compact interval I ^ R. Show that
for any e>0, there is a number S>0 such that for any increasing sequence *o =s£ t0 ^ xi ^ - • - ^ xk < 4 ^ xk+1 < • - • < xn of points of I for which xk+1 - xk ** 8, we have |
||
|
|
||
|
("Riemann sums"; consider first the case in which/is a step-function).
|
||
|
|
||